Retracts that are kernels of locally nilpotent derivations

Dayan Liu; Xiaosong Sun

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 1, page 191-199
  • ISSN: 0011-4642

Abstract

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Let k be a field of characteristic zero and B a k -domain. Let R be a retract of B being the kernel of a locally nilpotent derivation of B . We show that if B = R ⊕ I for some principal ideal I (in particular, if B is a UFD), then B = R [ 1 ] , i.e., B is a polynomial algebra over R in one variable. It is natural to ask that, if a retract R of a k -UFD B is the kernel of two commuting locally nilpotent derivations of B , then does it follow that B ≅ R [ 2 ] ? We give a negative answer to this question. The interest in retracts comes from the fact that they are closely related to Zariski’s cancellation problem and the Jacobian conjecture in affine algebraic geometry.

How to cite

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Liu, Dayan, and Sun, Xiaosong. "Retracts that are kernels of locally nilpotent derivations." Czechoslovak Mathematical Journal 72.1 (2022): 191-199. <http://eudml.org/doc/297785>.

@article{Liu2022,
abstract = {Let $k$ be a field of characteristic zero and $B$ a $k$-domain. Let $R$ be a retract of $B$ being the kernel of a locally nilpotent derivation of $B$. We show that if $B=R\oplus I$ for some principal ideal $I$ (in particular, if $B$ is a UFD), then $B= R^\{[1]\}$, i.e., $B$ is a polynomial algebra over $R$ in one variable. It is natural to ask that, if a retract $R$ of a $k$-UFD $B$ is the kernel of two commuting locally nilpotent derivations of $B$, then does it follow that $B\cong R^\{[2]\}$? We give a negative answer to this question. The interest in retracts comes from the fact that they are closely related to Zariski’s cancellation problem and the Jacobian conjecture in affine algebraic geometry.},
author = {Liu, Dayan, Sun, Xiaosong},
journal = {Czechoslovak Mathematical Journal},
keywords = {retract; locally nilpotent derivation; kernel; Zariski's cancellation problem},
language = {eng},
number = {1},
pages = {191-199},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Retracts that are kernels of locally nilpotent derivations},
url = {http://eudml.org/doc/297785},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Liu, Dayan
AU - Sun, Xiaosong
TI - Retracts that are kernels of locally nilpotent derivations
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 1
SP - 191
EP - 199
AB - Let $k$ be a field of characteristic zero and $B$ a $k$-domain. Let $R$ be a retract of $B$ being the kernel of a locally nilpotent derivation of $B$. We show that if $B=R\oplus I$ for some principal ideal $I$ (in particular, if $B$ is a UFD), then $B= R^{[1]}$, i.e., $B$ is a polynomial algebra over $R$ in one variable. It is natural to ask that, if a retract $R$ of a $k$-UFD $B$ is the kernel of two commuting locally nilpotent derivations of $B$, then does it follow that $B\cong R^{[2]}$? We give a negative answer to this question. The interest in retracts comes from the fact that they are closely related to Zariski’s cancellation problem and the Jacobian conjecture in affine algebraic geometry.
LA - eng
KW - retract; locally nilpotent derivation; kernel; Zariski's cancellation problem
UR - http://eudml.org/doc/297785
ER -

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